grandes-ecoles 2020 Q12

grandes-ecoles · France · centrale-maths1__pc Matrices Bilinear and Symplectic Form Properties
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$. 
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$.
Paper Questions
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